I do believe they have negative effects on children but I don't believe that extends to school shootings.
Are you sure it was a lack of video games and movies that let you to have trouble forming relationships? My anecdote is that even though I had virtually no video games (except at friends houses) or television until highschool I was still able to form (and maintain for 30+ years) friendships with my peers who had very different rules than I did, and had a full social life all through school (I do consider myself a strong introvert and have had social anxiety issues in past, but I would never point at lack of TV/games growing up as a root cause).
Like you said, neither one of us has hard data, only anecdotal stories with different outcomes for similar "media" parenting strategies.
Even if there is no negative consequence on an individual child relative to their peer group, I still believe (can't support with evidence obviously) that exposing children to violence in video games and movies is a net societal negative that I don't intend to contribute to.
edit: Just wanted to say I appreciate your perspective as it's one I hadn't even considered before and gave me some food for thought.